Pretty cool stuff: the PlayStation 4 apparently runs on a version of FreeBSD 9.0 modified by Sony and renamed to 'Orbis OS'. Second generation developer kits use GNU GRUB to boot, and allow you to boot into a graphical or a console mode. Not much else is known at this point, but it's a fun bit of information, and perhaps a boon to homebrew developer and hackers.

Around the release of Direct3D 8, people starting using Direct3D versions as a shorthand for the capabilities of graphics cards.

D3D 9 hardware, for example, is hardware that you could fully implement D3D 9 with. Or you could implement OpenGL 2.0, or some completely different API. The PS3 and Wii U both have D3D 9 hardware, but they both use their own proprietary APIs, while the Xbox 360 uses a modified version of D3D 9.

D3D 11 hardware is hardware you could implement D3D 11 on, or OpenGL 4.3. The Xbox One and PS4 have the same GPU, but Microsoft are using D3D 11, while Sony are using an updated version of their own API.

DirectX is owned by Microsoft. I presume it is very unlikely that they licensed a core technology of their Xbox to Sony.

The shaders in the graphics pipeline are designed to meet the Direct3D 11 specs, because they are based on standard Radeon parts. In a world where one could boot Windows on a PS4, DirectX stuff should run on it.